3.1 \(\int (e x)^m (a+b x^n)^3 (A+B x^n) (c+d x^n) \, dx\)

Optimal. Leaf size=210 \[ \frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {a^2 x^{n+1} (e x)^m (a A d+a B c+3 A b c)}{m+n+1}+\frac {b^2 x^{4 n+1} (e x)^m (3 a B d+A b d+b B c)}{m+4 n+1}+\frac {a x^{2 n+1} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac {b x^{3 n+1} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac {b^3 B d x^{5 n+1} (e x)^m}{m+5 n+1} \]

[Out]

a^2*(A*a*d+3*A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+a*(3*A*b*(a*d+b*c)+a*B*(a*d+3*b*c))*x^(1+2*n)*(e*x)^m/(1+m+2
*n)+b*(3*a*B*(a*d+b*c)+A*b*(3*a*d+b*c))*x^(1+3*n)*(e*x)^m/(1+m+3*n)+b^2*(A*b*d+3*B*a*d+B*b*c)*x^(1+4*n)*(e*x)^
m/(1+m+4*n)+b^3*B*d*x^(1+5*n)*(e*x)^m/(1+m+5*n)+a^3*A*c*(e*x)^(1+m)/e/(1+m)

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {570, 20, 30} \[ \frac {a^2 x^{n+1} (e x)^m (a A d+a B c+3 A b c)}{m+n+1}+\frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {b^2 x^{4 n+1} (e x)^m (3 a B d+A b d+b B c)}{m+4 n+1}+\frac {a x^{2 n+1} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac {b x^{3 n+1} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac {b^3 B d x^{5 n+1} (e x)^m}{m+5 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^n)^3*(A + B*x^n)*(c + d*x^n),x]

[Out]

(a^2*(3*A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*x^(
1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n
) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b^3*B*d*x^(1 + 5*n)*(e*x)^m)/(1 + m +
5*n) + (a^3*A*c*(e*x)^(1 + m))/(e*(1 + m))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin {align*} \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx &=\int \left (a^3 A c (e x)^m+a^2 (3 A b c+a B c+a A d) x^n (e x)^m+a (3 A b (b c+a d)+a B (3 b c+a d)) x^{2 n} (e x)^m+b (3 a B (b c+a d)+A b (b c+3 a d)) x^{3 n} (e x)^m+b^2 (b B c+A b d+3 a B d) x^{4 n} (e x)^m+b^3 B d x^{5 n} (e x)^m\right ) \, dx\\ &=\frac {a^3 A c (e x)^{1+m}}{e (1+m)}+\left (b^3 B d\right ) \int x^{5 n} (e x)^m \, dx+\left (a^2 (3 A b c+a B c+a A d)\right ) \int x^n (e x)^m \, dx+\left (b^2 (b B c+A b d+3 a B d)\right ) \int x^{4 n} (e x)^m \, dx+(a (3 A b (b c+a d)+a B (3 b c+a d))) \int x^{2 n} (e x)^m \, dx+(b (3 a B (b c+a d)+A b (b c+3 a d))) \int x^{3 n} (e x)^m \, dx\\ &=\frac {a^3 A c (e x)^{1+m}}{e (1+m)}+\left (b^3 B d x^{-m} (e x)^m\right ) \int x^{m+5 n} \, dx+\left (a^2 (3 A b c+a B c+a A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left (b^2 (b B c+A b d+3 a B d) x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (a (3 A b (b c+a d)+a B (3 b c+a d)) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx+\left (b (3 a B (b c+a d)+A b (b c+3 a d)) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx\\ &=\frac {a^2 (3 A b c+a B c+a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b^2 (b B c+A b d+3 a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^3 B d x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a^3 A c (e x)^{1+m}}{e (1+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.99, size = 172, normalized size = 0.82 \[ x (e x)^m \left (\frac {a^3 A c}{m+1}+\frac {a^2 x^n (a A d+a B c+3 A b c)}{m+n+1}+\frac {b^2 x^{4 n} (3 a B d+A b d+b B c)}{m+4 n+1}+\frac {a x^{2 n} (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac {b x^{3 n} (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac {b^3 B d x^{5 n}}{m+5 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^n)^3*(A + B*x^n)*(c + d*x^n),x]

[Out]

x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + (a*(3*A*b*(b*c + a*d) + a*B*(
3*b*c + a*d))*x^(2*n))/(1 + m + 2*n) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^(3*n))/(1 + m + 3*n) + (b^
2*(b*B*c + A*b*d + 3*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^3*B*d*x^(5*n))/(1 + m + 5*n))

________________________________________________________________________________________

fricas [B]  time = 0.81, size = 3073, normalized size = 14.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="fricas")

[Out]

((B*b^3*d*m^5 + 5*B*b^3*d*m^4 + 10*B*b^3*d*m^3 + 10*B*b^3*d*m^2 + 5*B*b^3*d*m + B*b^3*d + 24*(B*b^3*d*m + B*b^
3*d)*n^4 + 50*(B*b^3*d*m^2 + 2*B*b^3*d*m + B*b^3*d)*n^3 + 35*(B*b^3*d*m^3 + 3*B*b^3*d*m^2 + 3*B*b^3*d*m + B*b^
3*d)*n^2 + 10*(B*b^3*d*m^4 + 4*B*b^3*d*m^3 + 6*B*b^3*d*m^2 + 4*B*b^3*d*m + B*b^3*d)*n)*x*x^(5*n)*e^(m*log(e) +
m*log(x)) + ((B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^5 + B*b^3*c + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 30*(
B*b^3*c + (3*B*a*b^2 + A*b^3)*d + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n^4 + 10*(B*b^3*c + (3*B*a*b^2 + A*b^3)
*d)*m^3 + 61*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^3)*d + 2*(B*b^3*c + (3*B*a*b^
2 + A*b^3)*d)*m)*n^3 + 10*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + 41*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)
*d)*m^3 + 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^3)*d + 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d
)*m)*n^2 + (3*B*a*b^2 + A*b^3)*d + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m + 11*(B*b^3*c + (B*b^3*c + (3*B*a*b^2
+ A*b^3)*d)*m^4 + 4*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3 + 6*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*
b^2 + A*b^3)*d + 4*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n)*x*x^(4*n)*e^(m*log(e) + m*log(x)) + (((3*B*a*b^2 +
A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^5 + 5*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4 + 40*((3*B*a
*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + ((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^4 + 10*((3*
B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 78*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2
+ (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 2*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^3
+ 10*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + 49*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^
2)*d)*m^3 + 3*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a
*b^2)*d + 3*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*n^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*
a*b^2)*d + 5*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m + 12*(((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A
*a*b^2)*d)*m^4 + 4*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 6*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2
*b + A*a*b^2)*d)*m^2 + (3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d + 4*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b
+ A*a*b^2)*d)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^
5 + 5*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 60*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b
)*d + (3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n^4 + 10*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^
2*b)*d)*m^3 + 107*((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 +
3*A*a^2*b)*d + 2*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n^3 + 10*(3*(B*a^2*b + A*a*b^2)*c + (B*a
^3 + 3*A*a^2*b)*d)*m^2 + 59*((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^3 + 3*(3*(B*a^2*b + A*a*b^2)*
c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 3*(3*(B*a^2*b + A*a*b^2)*c
+ (B*a^3 + 3*A*a^2*b)*d)*m)*n^2 + 3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 5*(3*(B*a^2*b + A*a*b^2)*c
+ (B*a^3 + 3*A*a^2*b)*d)*m + 13*((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 4*(3*(B*a^2*b + A*a*
b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^3 + 6*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 3*(B*a^2*b + A
*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d + 4*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*n)*x*x^(2*n)*e^(m*l
og(e) + m*log(x)) + ((A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^5 + A*a^3*d + 5*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^4
+ 120*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n^4 + 10*(A*a^3*d + (B*a^3 + 3*
A*a^2*b)*c)*m^3 + 154*(A*a^3*d + (A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + (B*a^3 + 3*A*a^2*b)*c + 2*(A*a^3*d +
(B*a^3 + 3*A*a^2*b)*c)*m)*n^3 + 10*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + 71*(A*a^3*d + (A*a^3*d + (B*a^3 + 3
*A*a^2*b)*c)*m^3 + 3*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + (B*a^3 + 3*A*a^2*b)*c + 3*(A*a^3*d + (B*a^3 + 3*A
*a^2*b)*c)*m)*n^2 + (B*a^3 + 3*A*a^2*b)*c + 5*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m + 14*(A*a^3*d + (A*a^3*d + (
B*a^3 + 3*A*a^2*b)*c)*m^4 + 4*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^3 + 6*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2
+ (B*a^3 + 3*A*a^2*b)*c + 4*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*a^3*c*m
^5 + 120*A*a^3*c*n^5 + 5*A*a^3*c*m^4 + 10*A*a^3*c*m^3 + 10*A*a^3*c*m^2 + 5*A*a^3*c*m + A*a^3*c + 274*(A*a^3*c*
m + A*a^3*c)*n^4 + 225*(A*a^3*c*m^2 + 2*A*a^3*c*m + A*a^3*c)*n^3 + 85*(A*a^3*c*m^3 + 3*A*a^3*c*m^2 + 3*A*a^3*c
*m + A*a^3*c)*n^2 + 15*(A*a^3*c*m^4 + 4*A*a^3*c*m^3 + 6*A*a^3*c*m^2 + 4*A*a^3*c*m + A*a^3*c)*n)*x*e^(m*log(e)
+ m*log(x)))/(m^6 + 120*(m + 1)*n^5 + 6*m^5 + 274*(m^2 + 2*m + 1)*n^4 + 15*m^4 + 225*(m^3 + 3*m^2 + 3*m + 1)*n
^3 + 20*m^3 + 85*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n^2 + 15*m^2 + 15*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n
+ 6*m + 1)

________________________________________________________________________________________

giac [B]  time = 1.19, size = 6927, normalized size = 32.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="giac")

[Out]

(B*b^3*d*m^5*x*x^m*x^(5*n)*e^m + 10*B*b^3*d*m^4*n*x*x^m*x^(5*n)*e^m + 35*B*b^3*d*m^3*n^2*x*x^m*x^(5*n)*e^m + 5
0*B*b^3*d*m^2*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^3*d*m*n^4*x*x^m*x^(5*n)*e^m + B*b^3*c*m^5*x*x^m*x^(4*n)*e^m + 3*B
*a*b^2*d*m^5*x*x^m*x^(4*n)*e^m + A*b^3*d*m^5*x*x^m*x^(4*n)*e^m + 11*B*b^3*c*m^4*n*x*x^m*x^(4*n)*e^m + 33*B*a*b
^2*d*m^4*n*x*x^m*x^(4*n)*e^m + 11*A*b^3*d*m^4*n*x*x^m*x^(4*n)*e^m + 41*B*b^3*c*m^3*n^2*x*x^m*x^(4*n)*e^m + 123
*B*a*b^2*d*m^3*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^3*d*m^3*n^2*x*x^m*x^(4*n)*e^m + 61*B*b^3*c*m^2*n^3*x*x^m*x^(4*n)
*e^m + 183*B*a*b^2*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 61*A*b^3*d*m^2*n^3*x*x^m*x^(4*n)*e^m + 30*B*b^3*c*m*n^4*x*x^m
*x^(4*n)*e^m + 90*B*a*b^2*d*m*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^3*d*m*n^4*x*x^m*x^(4*n)*e^m + 3*B*a*b^2*c*m^5*x*x
^m*x^(3*n)*e^m + A*b^3*c*m^5*x*x^m*x^(3*n)*e^m + 3*B*a^2*b*d*m^5*x*x^m*x^(3*n)*e^m + 3*A*a*b^2*d*m^5*x*x^m*x^(
3*n)*e^m + 36*B*a*b^2*c*m^4*n*x*x^m*x^(3*n)*e^m + 12*A*b^3*c*m^4*n*x*x^m*x^(3*n)*e^m + 36*B*a^2*b*d*m^4*n*x*x^
m*x^(3*n)*e^m + 36*A*a*b^2*d*m^4*n*x*x^m*x^(3*n)*e^m + 147*B*a*b^2*c*m^3*n^2*x*x^m*x^(3*n)*e^m + 49*A*b^3*c*m^
3*n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*b*d*m^3*n^2*x*x^m*x^(3*n)*e^m + 147*A*a*b^2*d*m^3*n^2*x*x^m*x^(3*n)*e^m +
234*B*a*b^2*c*m^2*n^3*x*x^m*x^(3*n)*e^m + 78*A*b^3*c*m^2*n^3*x*x^m*x^(3*n)*e^m + 234*B*a^2*b*d*m^2*n^3*x*x^m*x
^(3*n)*e^m + 234*A*a*b^2*d*m^2*n^3*x*x^m*x^(3*n)*e^m + 120*B*a*b^2*c*m*n^4*x*x^m*x^(3*n)*e^m + 40*A*b^3*c*m*n^
4*x*x^m*x^(3*n)*e^m + 120*B*a^2*b*d*m*n^4*x*x^m*x^(3*n)*e^m + 120*A*a*b^2*d*m*n^4*x*x^m*x^(3*n)*e^m + 3*B*a^2*
b*c*m^5*x*x^m*x^(2*n)*e^m + 3*A*a*b^2*c*m^5*x*x^m*x^(2*n)*e^m + B*a^3*d*m^5*x*x^m*x^(2*n)*e^m + 3*A*a^2*b*d*m^
5*x*x^m*x^(2*n)*e^m + 39*B*a^2*b*c*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*a*b^2*c*m^4*n*x*x^m*x^(2*n)*e^m + 13*B*a^3*d
*m^4*n*x*x^m*x^(2*n)*e^m + 39*A*a^2*b*d*m^4*n*x*x^m*x^(2*n)*e^m + 177*B*a^2*b*c*m^3*n^2*x*x^m*x^(2*n)*e^m + 17
7*A*a*b^2*c*m^3*n^2*x*x^m*x^(2*n)*e^m + 59*B*a^3*d*m^3*n^2*x*x^m*x^(2*n)*e^m + 177*A*a^2*b*d*m^3*n^2*x*x^m*x^(
2*n)*e^m + 321*B*a^2*b*c*m^2*n^3*x*x^m*x^(2*n)*e^m + 321*A*a*b^2*c*m^2*n^3*x*x^m*x^(2*n)*e^m + 107*B*a^3*d*m^2
*n^3*x*x^m*x^(2*n)*e^m + 321*A*a^2*b*d*m^2*n^3*x*x^m*x^(2*n)*e^m + 180*B*a^2*b*c*m*n^4*x*x^m*x^(2*n)*e^m + 180
*A*a*b^2*c*m*n^4*x*x^m*x^(2*n)*e^m + 60*B*a^3*d*m*n^4*x*x^m*x^(2*n)*e^m + 180*A*a^2*b*d*m*n^4*x*x^m*x^(2*n)*e^
m + B*a^3*c*m^5*x*x^m*x^n*e^m + 3*A*a^2*b*c*m^5*x*x^m*x^n*e^m + A*a^3*d*m^5*x*x^m*x^n*e^m + 14*B*a^3*c*m^4*n*x
*x^m*x^n*e^m + 42*A*a^2*b*c*m^4*n*x*x^m*x^n*e^m + 14*A*a^3*d*m^4*n*x*x^m*x^n*e^m + 71*B*a^3*c*m^3*n^2*x*x^m*x^
n*e^m + 213*A*a^2*b*c*m^3*n^2*x*x^m*x^n*e^m + 71*A*a^3*d*m^3*n^2*x*x^m*x^n*e^m + 154*B*a^3*c*m^2*n^3*x*x^m*x^n
*e^m + 462*A*a^2*b*c*m^2*n^3*x*x^m*x^n*e^m + 154*A*a^3*d*m^2*n^3*x*x^m*x^n*e^m + 120*B*a^3*c*m*n^4*x*x^m*x^n*e
^m + 360*A*a^2*b*c*m*n^4*x*x^m*x^n*e^m + 120*A*a^3*d*m*n^4*x*x^m*x^n*e^m + A*a^3*c*m^5*x*x^m*e^m + 15*A*a^3*c*
m^4*n*x*x^m*e^m + 85*A*a^3*c*m^3*n^2*x*x^m*e^m + 225*A*a^3*c*m^2*n^3*x*x^m*e^m + 274*A*a^3*c*m*n^4*x*x^m*e^m +
120*A*a^3*c*n^5*x*x^m*e^m + 5*B*b^3*d*m^4*x*x^m*x^(5*n)*e^m + 40*B*b^3*d*m^3*n*x*x^m*x^(5*n)*e^m + 105*B*b^3*
d*m^2*n^2*x*x^m*x^(5*n)*e^m + 100*B*b^3*d*m*n^3*x*x^m*x^(5*n)*e^m + 24*B*b^3*d*n^4*x*x^m*x^(5*n)*e^m + 5*B*b^3
*c*m^4*x*x^m*x^(4*n)*e^m + 15*B*a*b^2*d*m^4*x*x^m*x^(4*n)*e^m + 5*A*b^3*d*m^4*x*x^m*x^(4*n)*e^m + 44*B*b^3*c*m
^3*n*x*x^m*x^(4*n)*e^m + 132*B*a*b^2*d*m^3*n*x*x^m*x^(4*n)*e^m + 44*A*b^3*d*m^3*n*x*x^m*x^(4*n)*e^m + 123*B*b^
3*c*m^2*n^2*x*x^m*x^(4*n)*e^m + 369*B*a*b^2*d*m^2*n^2*x*x^m*x^(4*n)*e^m + 123*A*b^3*d*m^2*n^2*x*x^m*x^(4*n)*e^
m + 122*B*b^3*c*m*n^3*x*x^m*x^(4*n)*e^m + 366*B*a*b^2*d*m*n^3*x*x^m*x^(4*n)*e^m + 122*A*b^3*d*m*n^3*x*x^m*x^(4
*n)*e^m + 30*B*b^3*c*n^4*x*x^m*x^(4*n)*e^m + 90*B*a*b^2*d*n^4*x*x^m*x^(4*n)*e^m + 30*A*b^3*d*n^4*x*x^m*x^(4*n)
*e^m + 15*B*a*b^2*c*m^4*x*x^m*x^(3*n)*e^m + 5*A*b^3*c*m^4*x*x^m*x^(3*n)*e^m + 15*B*a^2*b*d*m^4*x*x^m*x^(3*n)*e
^m + 15*A*a*b^2*d*m^4*x*x^m*x^(3*n)*e^m + 144*B*a*b^2*c*m^3*n*x*x^m*x^(3*n)*e^m + 48*A*b^3*c*m^3*n*x*x^m*x^(3*
n)*e^m + 144*B*a^2*b*d*m^3*n*x*x^m*x^(3*n)*e^m + 144*A*a*b^2*d*m^3*n*x*x^m*x^(3*n)*e^m + 441*B*a*b^2*c*m^2*n^2
*x*x^m*x^(3*n)*e^m + 147*A*b^3*c*m^2*n^2*x*x^m*x^(3*n)*e^m + 441*B*a^2*b*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 441*A*a
*b^2*d*m^2*n^2*x*x^m*x^(3*n)*e^m + 468*B*a*b^2*c*m*n^3*x*x^m*x^(3*n)*e^m + 156*A*b^3*c*m*n^3*x*x^m*x^(3*n)*e^m
+ 468*B*a^2*b*d*m*n^3*x*x^m*x^(3*n)*e^m + 468*A*a*b^2*d*m*n^3*x*x^m*x^(3*n)*e^m + 120*B*a*b^2*c*n^4*x*x^m*x^(
3*n)*e^m + 40*A*b^3*c*n^4*x*x^m*x^(3*n)*e^m + 120*B*a^2*b*d*n^4*x*x^m*x^(3*n)*e^m + 120*A*a*b^2*d*n^4*x*x^m*x^
(3*n)*e^m + 15*B*a^2*b*c*m^4*x*x^m*x^(2*n)*e^m + 15*A*a*b^2*c*m^4*x*x^m*x^(2*n)*e^m + 5*B*a^3*d*m^4*x*x^m*x^(2
*n)*e^m + 15*A*a^2*b*d*m^4*x*x^m*x^(2*n)*e^m + 156*B*a^2*b*c*m^3*n*x*x^m*x^(2*n)*e^m + 156*A*a*b^2*c*m^3*n*x*x
^m*x^(2*n)*e^m + 52*B*a^3*d*m^3*n*x*x^m*x^(2*n)*e^m + 156*A*a^2*b*d*m^3*n*x*x^m*x^(2*n)*e^m + 531*B*a^2*b*c*m^
2*n^2*x*x^m*x^(2*n)*e^m + 531*A*a*b^2*c*m^2*n^2*x*x^m*x^(2*n)*e^m + 177*B*a^3*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 53
1*A*a^2*b*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 642*B*a^2*b*c*m*n^3*x*x^m*x^(2*n)*e^m + 642*A*a*b^2*c*m*n^3*x*x^m*x^(2
*n)*e^m + 214*B*a^3*d*m*n^3*x*x^m*x^(2*n)*e^m + 642*A*a^2*b*d*m*n^3*x*x^m*x^(2*n)*e^m + 180*B*a^2*b*c*n^4*x*x^
m*x^(2*n)*e^m + 180*A*a*b^2*c*n^4*x*x^m*x^(2*n)*e^m + 60*B*a^3*d*n^4*x*x^m*x^(2*n)*e^m + 180*A*a^2*b*d*n^4*x*x
^m*x^(2*n)*e^m + 5*B*a^3*c*m^4*x*x^m*x^n*e^m + 15*A*a^2*b*c*m^4*x*x^m*x^n*e^m + 5*A*a^3*d*m^4*x*x^m*x^n*e^m +
56*B*a^3*c*m^3*n*x*x^m*x^n*e^m + 168*A*a^2*b*c*m^3*n*x*x^m*x^n*e^m + 56*A*a^3*d*m^3*n*x*x^m*x^n*e^m + 213*B*a^
3*c*m^2*n^2*x*x^m*x^n*e^m + 639*A*a^2*b*c*m^2*n^2*x*x^m*x^n*e^m + 213*A*a^3*d*m^2*n^2*x*x^m*x^n*e^m + 308*B*a^
3*c*m*n^3*x*x^m*x^n*e^m + 924*A*a^2*b*c*m*n^3*x*x^m*x^n*e^m + 308*A*a^3*d*m*n^3*x*x^m*x^n*e^m + 120*B*a^3*c*n^
4*x*x^m*x^n*e^m + 360*A*a^2*b*c*n^4*x*x^m*x^n*e^m + 120*A*a^3*d*n^4*x*x^m*x^n*e^m + 5*A*a^3*c*m^4*x*x^m*e^m +
60*A*a^3*c*m^3*n*x*x^m*e^m + 255*A*a^3*c*m^2*n^2*x*x^m*e^m + 450*A*a^3*c*m*n^3*x*x^m*e^m + 274*A*a^3*c*n^4*x*x
^m*e^m + 10*B*b^3*d*m^3*x*x^m*x^(5*n)*e^m + 60*B*b^3*d*m^2*n*x*x^m*x^(5*n)*e^m + 105*B*b^3*d*m*n^2*x*x^m*x^(5*
n)*e^m + 50*B*b^3*d*n^3*x*x^m*x^(5*n)*e^m + 10*B*b^3*c*m^3*x*x^m*x^(4*n)*e^m + 30*B*a*b^2*d*m^3*x*x^m*x^(4*n)*
e^m + 10*A*b^3*d*m^3*x*x^m*x^(4*n)*e^m + 66*B*b^3*c*m^2*n*x*x^m*x^(4*n)*e^m + 198*B*a*b^2*d*m^2*n*x*x^m*x^(4*n
)*e^m + 66*A*b^3*d*m^2*n*x*x^m*x^(4*n)*e^m + 123*B*b^3*c*m*n^2*x*x^m*x^(4*n)*e^m + 369*B*a*b^2*d*m*n^2*x*x^m*x
^(4*n)*e^m + 123*A*b^3*d*m*n^2*x*x^m*x^(4*n)*e^m + 61*B*b^3*c*n^3*x*x^m*x^(4*n)*e^m + 183*B*a*b^2*d*n^3*x*x^m*
x^(4*n)*e^m + 61*A*b^3*d*n^3*x*x^m*x^(4*n)*e^m + 30*B*a*b^2*c*m^3*x*x^m*x^(3*n)*e^m + 10*A*b^3*c*m^3*x*x^m*x^(
3*n)*e^m + 30*B*a^2*b*d*m^3*x*x^m*x^(3*n)*e^m + 30*A*a*b^2*d*m^3*x*x^m*x^(3*n)*e^m + 216*B*a*b^2*c*m^2*n*x*x^m
*x^(3*n)*e^m + 72*A*b^3*c*m^2*n*x*x^m*x^(3*n)*e^m + 216*B*a^2*b*d*m^2*n*x*x^m*x^(3*n)*e^m + 216*A*a*b^2*d*m^2*
n*x*x^m*x^(3*n)*e^m + 441*B*a*b^2*c*m*n^2*x*x^m*x^(3*n)*e^m + 147*A*b^3*c*m*n^2*x*x^m*x^(3*n)*e^m + 441*B*a^2*
b*d*m*n^2*x*x^m*x^(3*n)*e^m + 441*A*a*b^2*d*m*n^2*x*x^m*x^(3*n)*e^m + 234*B*a*b^2*c*n^3*x*x^m*x^(3*n)*e^m + 78
*A*b^3*c*n^3*x*x^m*x^(3*n)*e^m + 234*B*a^2*b*d*n^3*x*x^m*x^(3*n)*e^m + 234*A*a*b^2*d*n^3*x*x^m*x^(3*n)*e^m + 3
0*B*a^2*b*c*m^3*x*x^m*x^(2*n)*e^m + 30*A*a*b^2*c*m^3*x*x^m*x^(2*n)*e^m + 10*B*a^3*d*m^3*x*x^m*x^(2*n)*e^m + 30
*A*a^2*b*d*m^3*x*x^m*x^(2*n)*e^m + 234*B*a^2*b*c*m^2*n*x*x^m*x^(2*n)*e^m + 234*A*a*b^2*c*m^2*n*x*x^m*x^(2*n)*e
^m + 78*B*a^3*d*m^2*n*x*x^m*x^(2*n)*e^m + 234*A*a^2*b*d*m^2*n*x*x^m*x^(2*n)*e^m + 531*B*a^2*b*c*m*n^2*x*x^m*x^
(2*n)*e^m + 531*A*a*b^2*c*m*n^2*x*x^m*x^(2*n)*e^m + 177*B*a^3*d*m*n^2*x*x^m*x^(2*n)*e^m + 531*A*a^2*b*d*m*n^2*
x*x^m*x^(2*n)*e^m + 321*B*a^2*b*c*n^3*x*x^m*x^(2*n)*e^m + 321*A*a*b^2*c*n^3*x*x^m*x^(2*n)*e^m + 107*B*a^3*d*n^
3*x*x^m*x^(2*n)*e^m + 321*A*a^2*b*d*n^3*x*x^m*x^(2*n)*e^m + 10*B*a^3*c*m^3*x*x^m*x^n*e^m + 30*A*a^2*b*c*m^3*x*
x^m*x^n*e^m + 10*A*a^3*d*m^3*x*x^m*x^n*e^m + 84*B*a^3*c*m^2*n*x*x^m*x^n*e^m + 252*A*a^2*b*c*m^2*n*x*x^m*x^n*e^
m + 84*A*a^3*d*m^2*n*x*x^m*x^n*e^m + 213*B*a^3*c*m*n^2*x*x^m*x^n*e^m + 639*A*a^2*b*c*m*n^2*x*x^m*x^n*e^m + 213
*A*a^3*d*m*n^2*x*x^m*x^n*e^m + 154*B*a^3*c*n^3*x*x^m*x^n*e^m + 462*A*a^2*b*c*n^3*x*x^m*x^n*e^m + 154*A*a^3*d*n
^3*x*x^m*x^n*e^m + 10*A*a^3*c*m^3*x*x^m*e^m + 90*A*a^3*c*m^2*n*x*x^m*e^m + 255*A*a^3*c*m*n^2*x*x^m*e^m + 225*A
*a^3*c*n^3*x*x^m*e^m + 10*B*b^3*d*m^2*x*x^m*x^(5*n)*e^m + 40*B*b^3*d*m*n*x*x^m*x^(5*n)*e^m + 35*B*b^3*d*n^2*x*
x^m*x^(5*n)*e^m + 10*B*b^3*c*m^2*x*x^m*x^(4*n)*e^m + 30*B*a*b^2*d*m^2*x*x^m*x^(4*n)*e^m + 10*A*b^3*d*m^2*x*x^m
*x^(4*n)*e^m + 44*B*b^3*c*m*n*x*x^m*x^(4*n)*e^m + 132*B*a*b^2*d*m*n*x*x^m*x^(4*n)*e^m + 44*A*b^3*d*m*n*x*x^m*x
^(4*n)*e^m + 41*B*b^3*c*n^2*x*x^m*x^(4*n)*e^m + 123*B*a*b^2*d*n^2*x*x^m*x^(4*n)*e^m + 41*A*b^3*d*n^2*x*x^m*x^(
4*n)*e^m + 30*B*a*b^2*c*m^2*x*x^m*x^(3*n)*e^m + 10*A*b^3*c*m^2*x*x^m*x^(3*n)*e^m + 30*B*a^2*b*d*m^2*x*x^m*x^(3
*n)*e^m + 30*A*a*b^2*d*m^2*x*x^m*x^(3*n)*e^m + 144*B*a*b^2*c*m*n*x*x^m*x^(3*n)*e^m + 48*A*b^3*c*m*n*x*x^m*x^(3
*n)*e^m + 144*B*a^2*b*d*m*n*x*x^m*x^(3*n)*e^m + 144*A*a*b^2*d*m*n*x*x^m*x^(3*n)*e^m + 147*B*a*b^2*c*n^2*x*x^m*
x^(3*n)*e^m + 49*A*b^3*c*n^2*x*x^m*x^(3*n)*e^m + 147*B*a^2*b*d*n^2*x*x^m*x^(3*n)*e^m + 147*A*a*b^2*d*n^2*x*x^m
*x^(3*n)*e^m + 30*B*a^2*b*c*m^2*x*x^m*x^(2*n)*e^m + 30*A*a*b^2*c*m^2*x*x^m*x^(2*n)*e^m + 10*B*a^3*d*m^2*x*x^m*
x^(2*n)*e^m + 30*A*a^2*b*d*m^2*x*x^m*x^(2*n)*e^m + 156*B*a^2*b*c*m*n*x*x^m*x^(2*n)*e^m + 156*A*a*b^2*c*m*n*x*x
^m*x^(2*n)*e^m + 52*B*a^3*d*m*n*x*x^m*x^(2*n)*e^m + 156*A*a^2*b*d*m*n*x*x^m*x^(2*n)*e^m + 177*B*a^2*b*c*n^2*x*
x^m*x^(2*n)*e^m + 177*A*a*b^2*c*n^2*x*x^m*x^(2*n)*e^m + 59*B*a^3*d*n^2*x*x^m*x^(2*n)*e^m + 177*A*a^2*b*d*n^2*x
*x^m*x^(2*n)*e^m + 10*B*a^3*c*m^2*x*x^m*x^n*e^m + 30*A*a^2*b*c*m^2*x*x^m*x^n*e^m + 10*A*a^3*d*m^2*x*x^m*x^n*e^
m + 56*B*a^3*c*m*n*x*x^m*x^n*e^m + 168*A*a^2*b*c*m*n*x*x^m*x^n*e^m + 56*A*a^3*d*m*n*x*x^m*x^n*e^m + 71*B*a^3*c
*n^2*x*x^m*x^n*e^m + 213*A*a^2*b*c*n^2*x*x^m*x^n*e^m + 71*A*a^3*d*n^2*x*x^m*x^n*e^m + 10*A*a^3*c*m^2*x*x^m*e^m
+ 60*A*a^3*c*m*n*x*x^m*e^m + 85*A*a^3*c*n^2*x*x^m*e^m + 5*B*b^3*d*m*x*x^m*x^(5*n)*e^m + 10*B*b^3*d*n*x*x^m*x^
(5*n)*e^m + 5*B*b^3*c*m*x*x^m*x^(4*n)*e^m + 15*B*a*b^2*d*m*x*x^m*x^(4*n)*e^m + 5*A*b^3*d*m*x*x^m*x^(4*n)*e^m +
11*B*b^3*c*n*x*x^m*x^(4*n)*e^m + 33*B*a*b^2*d*n*x*x^m*x^(4*n)*e^m + 11*A*b^3*d*n*x*x^m*x^(4*n)*e^m + 15*B*a*b
^2*c*m*x*x^m*x^(3*n)*e^m + 5*A*b^3*c*m*x*x^m*x^(3*n)*e^m + 15*B*a^2*b*d*m*x*x^m*x^(3*n)*e^m + 15*A*a*b^2*d*m*x
*x^m*x^(3*n)*e^m + 36*B*a*b^2*c*n*x*x^m*x^(3*n)*e^m + 12*A*b^3*c*n*x*x^m*x^(3*n)*e^m + 36*B*a^2*b*d*n*x*x^m*x^
(3*n)*e^m + 36*A*a*b^2*d*n*x*x^m*x^(3*n)*e^m + 15*B*a^2*b*c*m*x*x^m*x^(2*n)*e^m + 15*A*a*b^2*c*m*x*x^m*x^(2*n)
*e^m + 5*B*a^3*d*m*x*x^m*x^(2*n)*e^m + 15*A*a^2*b*d*m*x*x^m*x^(2*n)*e^m + 39*B*a^2*b*c*n*x*x^m*x^(2*n)*e^m + 3
9*A*a*b^2*c*n*x*x^m*x^(2*n)*e^m + 13*B*a^3*d*n*x*x^m*x^(2*n)*e^m + 39*A*a^2*b*d*n*x*x^m*x^(2*n)*e^m + 5*B*a^3*
c*m*x*x^m*x^n*e^m + 15*A*a^2*b*c*m*x*x^m*x^n*e^m + 5*A*a^3*d*m*x*x^m*x^n*e^m + 14*B*a^3*c*n*x*x^m*x^n*e^m + 42
*A*a^2*b*c*n*x*x^m*x^n*e^m + 14*A*a^3*d*n*x*x^m*x^n*e^m + 5*A*a^3*c*m*x*x^m*e^m + 15*A*a^3*c*n*x*x^m*e^m + B*b
^3*d*x*x^m*x^(5*n)*e^m + B*b^3*c*x*x^m*x^(4*n)*e^m + 3*B*a*b^2*d*x*x^m*x^(4*n)*e^m + A*b^3*d*x*x^m*x^(4*n)*e^m
+ 3*B*a*b^2*c*x*x^m*x^(3*n)*e^m + A*b^3*c*x*x^m*x^(3*n)*e^m + 3*B*a^2*b*d*x*x^m*x^(3*n)*e^m + 3*A*a*b^2*d*x*x
^m*x^(3*n)*e^m + 3*B*a^2*b*c*x*x^m*x^(2*n)*e^m + 3*A*a*b^2*c*x*x^m*x^(2*n)*e^m + B*a^3*d*x*x^m*x^(2*n)*e^m + 3
*A*a^2*b*d*x*x^m*x^(2*n)*e^m + B*a^3*c*x*x^m*x^n*e^m + 3*A*a^2*b*c*x*x^m*x^n*e^m + A*a^3*d*x*x^m*x^n*e^m + A*a
^3*c*x*x^m*e^m)/(m^6 + 15*m^5*n + 85*m^4*n^2 + 225*m^3*n^3 + 274*m^2*n^4 + 120*m*n^5 + 6*m^5 + 75*m^4*n + 340*
m^3*n^2 + 675*m^2*n^3 + 548*m*n^4 + 120*n^5 + 15*m^4 + 150*m^3*n + 510*m^2*n^2 + 675*m*n^3 + 274*n^4 + 20*m^3
+ 150*m^2*n + 340*m*n^2 + 225*n^3 + 15*m^2 + 75*m*n + 85*n^2 + 6*m + 15*n + 1)

________________________________________________________________________________________

maple [C]  time = 0.21, size = 4972, normalized size = 23.68 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^n+a)^3*(A+B*x^n)*(d*x^n+c),x)

[Out]

x*(41*A*b^3*d*m^3*n^2*(x^n)^4+61*A*b^3*d*m^2*n^3*(x^n)^4+30*A*b^3*d*m*n^4*(x^n)^4+24*B*b^3*d*m*n^4*(x^n)^5+11*
A*b^3*d*m^4*n*(x^n)^4+40*B*b^3*d*m^3*n*(x^n)^5+3*B*a*b^2*d*m^5*(x^n)^4+11*B*b^3*c*m^4*n*(x^n)^4+41*B*b^3*c*m^3
*n^2*(x^n)^4+61*B*b^3*c*m^2*n^3*(x^n)^4+10*B*b^3*d*m^4*n*(x^n)^5+35*B*b^3*d*m^3*n^2*(x^n)^5+50*B*b^3*d*m^2*n^3
*(x^n)^5+105*B*b^3*d*m^2*n^2*(x^n)^5+100*B*b^3*d*m*n^3*(x^n)^5+3*A*a*b^2*d*m^5*(x^n)^3+12*A*b^3*c*m^4*n*(x^n)^
3+49*A*b^3*c*m^3*n^2*(x^n)^3+78*A*b^3*c*m^2*n^3*(x^n)^3+40*A*b^3*c*m*n^4*(x^n)^3+44*A*b^3*d*m^3*n*(x^n)^4+123*
A*b^3*d*m^2*n^2*(x^n)^4+30*B*b^3*c*m*n^4*(x^n)^4+3*B*a*b^2*c*m^5*(x^n)^3+15*B*a*b^2*d*m^4*(x^n)^4+90*B*a*b^2*d
*n^4*(x^n)^4+44*B*b^3*c*m^3*n*(x^n)^4+123*B*b^3*c*m^2*n^2*(x^n)^4+122*B*b^3*c*m*n^3*(x^n)^4+60*B*b^3*d*m^2*n*(
x^n)^5+105*B*b^3*d*m*n^2*(x^n)^5+3*A*a^2*b*d*m^5*(x^n)^2+3*A*a*b^2*c*m^5*(x^n)^2+66*A*b^3*d*m^2*n*(x^n)^4+123*
A*b^3*d*m*n^2*(x^n)^4+122*A*b^3*d*m*n^3*(x^n)^4+3*B*a^2*b*d*m^5*(x^n)^3+15*A*a*b^2*d*m^4*(x^n)^3+120*A*a*b^2*d
*n^4*(x^n)^3+48*A*b^3*c*m^3*n*(x^n)^3+147*A*b^3*c*m^2*n^2*(x^n)^3+156*A*b^3*c*m*n^3*(x^n)^3+15*B*a*b^2*c*m^4*(
x^n)^3+120*B*a*b^2*c*n^4*(x^n)^3+30*B*a*b^2*d*m^3*(x^n)^4+183*B*a*b^2*d*n^3*(x^n)^4+66*B*b^3*c*m^2*n*(x^n)^4+1
23*B*b^3*c*m*n^2*(x^n)^4+40*B*b^3*d*m*n*(x^n)^5+14*A*a^3*d*m^4*n*x^n+71*A*a^3*d*m^3*n^2*x^n+120*B*a^3*c*m*n^4*
x^n+52*B*a^3*d*m^3*n*(x^n)^2+13*B*a^3*d*m^4*n*(x^n)^2+59*B*a^3*d*m^3*n^2*(x^n)^2+107*B*a^3*d*m^2*n^3*(x^n)^2+6
0*B*a^3*d*m*n^4*(x^n)^2+3*B*a^2*b*c*m^5*(x^n)^2+15*B*a^2*b*d*m^4*(x^n)^3+120*B*a^2*b*d*n^4*(x^n)^3+180*A*a*b^2
*c*n^4*(x^n)^2+30*A*a*b^2*d*m^3*(x^n)^3+234*A*a*b^2*d*n^3*(x^n)^3+72*A*b^3*c*m^2*n*(x^n)^3+147*A*b^3*c*m*n^2*(
x^n)^3+44*A*b^3*d*m*n*(x^n)^4+14*B*a^3*c*m^4*n*x^n+71*B*a^3*c*m^3*n^2*x^n+154*B*a^3*c*m^2*n^3*x^n+321*A*a^2*b*
d*n^3*(x^n)^2+30*A*a*b^2*c*m^3*(x^n)^2+321*A*a*b^2*c*n^3*(x^n)^2+154*A*a^3*d*m^2*n^3*x^n+120*A*a^3*d*m*n^4*x^n
+3*A*a^2*b*c*m^5*x^n+15*A*a^2*b*d*m^4*(x^n)^2+180*A*a^2*b*d*n^4*(x^n)^2+15*A*a*b^2*c*m^4*(x^n)^2+234*B*a^2*b*d
*n^3*(x^n)^3+30*B*a*b^2*c*m^3*(x^n)^3+234*B*a*b^2*c*n^3*(x^n)^3+30*B*a*b^2*d*m^2*(x^n)^4+123*B*a*b^2*d*n^2*(x^
n)^4+44*B*b^3*c*m*n*(x^n)^4+56*A*a^3*d*m^3*n*x^n+213*A*a^3*d*m^2*n^2*x^n+308*A*a^3*d*m*n^3*x^n+15*A*a^2*b*c*m^
4*x^n+360*A*a^2*b*c*n^4*x^n+30*A*a^2*b*d*m^3*(x^n)^2+84*A*a^3*d*m^2*n*x^n+213*A*a^3*d*m*n^2*x^n+30*A*a^2*b*c*m
^3*x^n+462*A*a^2*b*c*n^3*x^n+177*B*a^3*d*m^2*n^2*(x^n)^2+214*B*a^3*d*m*n^3*(x^n)^2+15*B*a^2*b*c*m^4*(x^n)^2+18
0*B*a^2*b*c*n^4*(x^n)^2+30*B*a^2*b*d*m^3*(x^n)^3+177*B*a^3*d*m*n^2*(x^n)^2+30*B*a^2*b*c*m^3*(x^n)^2+321*B*a^2*
b*c*n^3*(x^n)^2+30*B*a^2*b*d*m^2*(x^n)^3+147*B*a^2*b*d*n^2*(x^n)^3+30*B*a*b^2*c*m^2*(x^n)^3+147*B*a*b^2*c*n^2*
(x^n)^3+15*B*a*b^2*d*(x^n)^4*m+33*B*a*b^2*d*(x^n)^4*n+30*A*a*b^2*d*m^2*(x^n)^3+147*A*a*b^2*d*n^2*(x^n)^3+48*A*
b^3*c*m*n*(x^n)^3+56*B*a^3*c*m^3*n*x^n+213*B*a^3*c*m^2*n^2*x^n+308*B*a^3*c*m*n^3*x^n+78*B*a^3*d*m^2*n*(x^n)^2+
213*B*a^3*c*m*n^2*x^n+52*B*a^3*d*m*n*(x^n)^2+30*B*a^2*b*c*m^2*(x^n)^2+177*B*a^2*b*c*n^2*(x^n)^2+15*B*a^2*b*d*(
x^n)^3*m+36*B*a^2*b*d*(x^n)^3*n+15*B*a*b^2*c*(x^n)^3*m+30*A*a^2*b*d*m^2*(x^n)^2+177*A*a^2*b*d*n^2*(x^n)^2+30*A
*a*b^2*c*m^2*(x^n)^2+177*A*a*b^2*c*n^2*(x^n)^2+15*A*a*b^2*d*(x^n)^3*m+36*A*a*b^2*d*(x^n)^3*n+84*B*a^3*c*m^2*n*
x^n+30*A*a^2*b*c*m^2*x^n+213*A*a^2*b*c*n^2*x^n+15*A*a^2*b*d*(x^n)^2*m+39*A*a^2*b*d*(x^n)^2*n+15*A*a*b^2*c*(x^n
)^2*m+39*A*a*b^2*c*(x^n)^2*n+42*A*a^2*b*c*x^n*n+36*B*a*b^2*c*(x^n)^3*n+56*A*a^3*d*m*n*x^n+56*B*a^3*c*m*n*x^n+1
5*B*a^2*b*c*(x^n)^2*m+39*B*a^2*b*c*(x^n)^2*n+15*A*a^2*b*c*x^n*m+B*b^3*c*(x^n)^4+A*b^3*c*(x^n)^3+B*a^3*d*(x^n)^
2+A*a^3*d*x^n+B*a^3*c*x^n+b^3*B*d*(x^n)^5+A*b^3*d*(x^n)^4+10*A*a^3*c*m^3+225*A*a^3*c*n^3+10*A*a^3*c*m^2+85*A*a
^3*c*n^2+A*a^3*c*m^5+5*A*a^3*c*m^4+274*A*a^3*c*n^4+120*A*a^3*c*n^5+5*a^3*A*c*m+15*a^3*A*c*n+90*B*a*b^2*d*m*n^4
*(x^n)^4+36*A*a*b^2*d*m^4*n*(x^n)^3+147*A*a*b^2*d*m^3*n^2*(x^n)^3+234*A*a*b^2*d*m^2*n^3*(x^n)^3+a^3*A*c+468*B*
a^2*b*d*m*n^3*(x^n)^3+144*A*a*b^2*d*m*n*(x^n)^3+234*B*a^2*b*c*m^2*n*(x^n)^2+531*B*a^2*b*c*m*n^2*(x^n)^2+144*B*
a^2*b*d*m*n*(x^n)^3+144*B*a*b^2*c*m*n*(x^n)^3+3*(x^n)^3*B*a^2*b*d+3*(x^n)^4*B*a*b^2*d+156*A*a^2*b*d*m^3*n*(x^n
)^2+531*A*a^2*b*d*m^2*n^2*(x^n)^2+642*A*a^2*b*d*m*n^3*(x^n)^2+156*A*a*b^2*c*m^3*n*(x^n)^2+531*A*a*b^2*c*m^2*n^
2*(x^n)^2+642*A*a*b^2*c*m*n^3*(x^n)^2+216*A*a*b^2*d*m^2*n*(x^n)^3+441*A*a*b^2*d*m*n^2*(x^n)^3+252*A*a^2*b*c*m^
2*n*x^n+639*A*a^2*b*c*m*n^2*x^n+156*A*a^2*b*d*m*n*(x^n)^2+156*A*a*b^2*c*m*n*(x^n)^2+156*B*a^2*b*c*m^3*n*(x^n)^
2+531*B*a^2*b*c*m^2*n^2*(x^n)^2+642*B*a^2*b*c*m*n^3*(x^n)^2+216*B*a^2*b*d*m^2*n*(x^n)^3+369*B*a*b^2*d*m^2*n^2*
(x^n)^4+366*B*a*b^2*d*m*n^3*(x^n)^4+39*A*a^2*b*d*m^4*n*(x^n)^2+177*A*a^2*b*d*m^3*n^2*(x^n)^2+321*A*a^2*b*d*m^2
*n^3*(x^n)^2+321*B*a^2*b*c*m^2*n^3*(x^n)^2+180*B*a^2*b*c*m*n^4*(x^n)^2+144*B*a^2*b*d*m^3*n*(x^n)^3+441*B*a^2*b
*d*m^2*n^2*(x^n)^3+33*B*a*b^2*d*m^4*n*(x^n)^4+123*B*a*b^2*d*m^3*n^2*(x^n)^4+183*B*a*b^2*d*m^2*n^3*(x^n)^4+42*A
*a^2*b*c*m^4*n*x^n+213*A*a^2*b*c*m^3*n^2*x^n+462*A*a^2*b*c*m^2*n^3*x^n+360*A*a^2*b*c*m*n^4*x^n+639*A*a^2*b*c*m
^2*n^2*x^n+924*A*a^2*b*c*m*n^3*x^n+234*A*a^2*b*d*m^2*n*(x^n)^2+531*A*a^2*b*d*m*n^2*(x^n)^2+234*A*a*b^2*c*m^2*n
*(x^n)^2+531*A*a*b^2*c*m*n^2*(x^n)^2+120*A*a*b^2*d*m*n^4*(x^n)^3+36*B*a^2*b*d*m^4*n*(x^n)^3+147*B*a^2*b*d*m^3*
n^2*(x^n)^3+180*A*a^2*b*d*m*n^4*(x^n)^2+39*A*a*b^2*c*m^4*n*(x^n)^2+177*A*a*b^2*c*m^3*n^2*(x^n)^2+321*A*a*b^2*c
*m^2*n^3*(x^n)^2+180*A*a*b^2*c*m*n^4*(x^n)^2+144*A*a*b^2*d*m^3*n*(x^n)^3+441*A*a*b^2*d*m^2*n^2*(x^n)^3+468*A*a
*b^2*d*m*n^3*(x^n)^3+39*B*a^2*b*c*m^4*n*(x^n)^2+177*B*a^2*b*c*m^3*n^2*(x^n)^2+156*B*a^2*b*c*m*n*(x^n)^2+168*A*
a^2*b*c*m*n*x^n+234*B*a^2*b*d*m^2*n^3*(x^n)^3+120*B*a^2*b*d*m*n^4*(x^n)^3+36*B*a*b^2*c*m^4*n*(x^n)^3+147*B*a*b
^2*c*m^3*n^2*(x^n)^3+234*B*a*b^2*c*m^2*n^3*(x^n)^3+120*B*a*b^2*c*m*n^4*(x^n)^3+132*B*a*b^2*d*m^3*n*(x^n)^4+144
*B*a*b^2*c*m^3*n*(x^n)^3+441*B*a*b^2*c*m^2*n^2*(x^n)^3+468*B*a*b^2*c*m*n^3*(x^n)^3+198*B*a*b^2*d*m^2*n*(x^n)^4
+369*B*a*b^2*d*m*n^2*(x^n)^4+441*B*a^2*b*d*m*n^2*(x^n)^3+216*B*a*b^2*c*m^2*n*(x^n)^3+441*B*a*b^2*c*m*n^2*(x^n)
^3+132*B*a*b^2*d*m*n*(x^n)^4+168*A*a^2*b*c*m^3*n*x^n+107*B*a^3*d*n^3*(x^n)^2+5*B*b^3*c*(x^n)^4*m+3*(x^n)^2*d*a
^2*b*A+3*(x^n)^2*c*a*b^2*A+3*(x^n)^3*A*a*b^2*d+10*B*b^3*d*m^3*(x^n)^5+50*B*b^3*d*n^3*(x^n)^5+5*A*b^3*c*m^4*(x^
n)^3+40*A*b^3*c*n^4*(x^n)^3+10*A*b^3*d*m^3*(x^n)^4+61*A*b^3*d*n^3*(x^n)^4+B*a^3*d*m^5*(x^n)^2+10*B*b^3*c*m^3*(
x^n)^4+61*B*b^3*c*n^3*(x^n)^4+10*B*b^3*d*m^2*(x^n)^5+35*B*b^3*d*n^2*(x^n)^5+A*a^3*d*m^5*x^n+5*B*a^3*c*m^4*x^n+
120*B*a^3*c*n^4*x^n+10*A*a^3*d*m^3*x^n+3*(x^n)^2*c*a^2*b*B+3*x^n*c*a^2*b*A+3*(x^n)^3*B*a*b^2*c+10*B*a^3*c*m^3*
x^n+154*B*a^3*c*n^3*x^n+10*B*a^3*d*m^2*(x^n)^2+59*B*a^3*d*n^2*(x^n)^2+10*A*a^3*d*m^2*x^n+71*A*a^3*d*n^2*x^n+10
*B*a^3*c*m^2*x^n+71*B*a^3*c*n^2*x^n+5*B*a^3*d*(x^n)^2*m+13*B*a^3*d*(x^n)^2*n+B*b^3*d*m^5*(x^n)^5+11*B*b^3*c*(x
^n)^4*n+154*A*a^3*d*n^3*x^n+450*A*a^3*c*m*n^3+90*A*a^3*c*m^2*n+255*A*a^3*c*m*n^2+60*A*a^3*c*m*n+274*A*a^3*c*m*
n^4+60*A*a^3*c*m^3*n+255*A*a^3*c*m^2*n^2+15*A*a^3*c*m^4*n+85*A*a^3*c*m^3*n^2+225*A*a^3*c*m^2*n^3+A*b^3*d*m^5*(
x^n)^4+B*b^3*c*m^5*(x^n)^4+5*B*b^3*d*m^4*(x^n)^5+24*B*b^3*d*n^4*(x^n)^5+A*b^3*c*m^5*(x^n)^3+5*A*b^3*d*m^4*(x^n
)^4+30*A*b^3*d*n^4*(x^n)^4+5*B*b^3*c*m^4*(x^n)^4+30*B*b^3*c*n^4*(x^n)^4+5*B*a^3*c*x^n*m+14*B*a^3*c*x^n*n+10*A*
b^3*c*m^3*(x^n)^3+78*A*b^3*c*n^3*(x^n)^3+10*A*b^3*d*m^2*(x^n)^4+41*A*b^3*d*n^2*(x^n)^4+B*a^3*c*m^5*x^n+5*B*a^3
*d*m^4*(x^n)^2+60*B*a^3*d*n^4*(x^n)^2+10*B*b^3*c*m^2*(x^n)^4+120*A*a^3*d*n^4*x^n+10*A*b^3*c*m^2*(x^n)^3+49*A*b
^3*c*n^2*(x^n)^3+5*A*b^3*d*(x^n)^4*m+11*A*b^3*d*(x^n)^4*n+5*A*b^3*c*(x^n)^3*m+12*A*b^3*c*(x^n)^3*n+41*B*b^3*c*
n^2*(x^n)^4+5*m*b^3*B*d*(x^n)^5+10*b^3*B*d*(x^n)^5*n+5*A*a^3*d*m^4*x^n+10*B*a^3*d*m^3*(x^n)^2+5*A*a^3*d*x^n*m+
14*A*a^3*d*x^n*n)/(m+1)/(m+n+1)/(m+2*n+1)/(m+3*n+1)/(1+m+4*n)/(1+m+5*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)^3+I*Pi*cs
gn(I*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

________________________________________________________________________________________

maxima [B]  time = 0.94, size = 464, normalized size = 2.21 \[ \frac {B b^{3} d e^{m} x e^{\left (m \log \relax (x) + 5 \, n \log \relax (x)\right )}}{m + 5 \, n + 1} + \frac {B b^{3} c e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} d e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {A b^{3} d e^{m} x e^{\left (m \log \relax (x) + 4 \, n \log \relax (x)\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} c e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {A b^{3} c e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b d e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, A a b^{2} d e^{m} x e^{\left (m \log \relax (x) + 3 \, n \log \relax (x)\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b c e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, A a b^{2} c e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {B a^{3} d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {3 \, A a^{2} b d e^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )}}{m + 2 \, n + 1} + \frac {B a^{3} c e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {3 \, A a^{2} b c e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {A a^{3} d e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a^{3} c}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^3*(A+B*x^n)*(c+d*x^n),x, algorithm="maxima")

[Out]

B*b^3*d*e^m*x*e^(m*log(x) + 5*n*log(x))/(m + 5*n + 1) + B*b^3*c*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1)
+ 3*B*a*b^2*d*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + A*b^3*d*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n
+ 1) + 3*B*a*b^2*c*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + A*b^3*c*e^m*x*e^(m*log(x) + 3*n*log(x))/(m
+ 3*n + 1) + 3*B*a^2*b*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 3*A*a*b^2*d*e^m*x*e^(m*log(x) + 3*n*
log(x))/(m + 3*n + 1) + 3*B*a^2*b*c*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a*b^2*c*e^m*x*e^(m*log
(x) + 2*n*log(x))/(m + 2*n + 1) + B*a^3*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a^2*b*d*e^m*x*e^
(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*a^3*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 3*A*a^2*b*c*e^m*x*
e^(m*log(x) + n*log(x))/(m + n + 1) + A*a^3*d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*a^3*
c/(e*(m + 1))

________________________________________________________________________________________

mupad [B]  time = 5.64, size = 1089, normalized size = 5.19 \[ \frac {A\,a^3\,c\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {b^2\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+B\,b\,c\right )\,\left (m^4+11\,m^3\,n+4\,m^3+41\,m^2\,n^2+33\,m^2\,n+6\,m^2+61\,m\,n^3+82\,m\,n^2+33\,m\,n+4\,m+30\,n^4+61\,n^3+41\,n^2+11\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (3\,A\,b^2\,c+B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^4+13\,m^3\,n+4\,m^3+59\,m^2\,n^2+39\,m^2\,n+6\,m^2+107\,m\,n^3+118\,m\,n^2+39\,m\,n+4\,m+60\,n^4+107\,n^3+59\,n^2+13\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {b\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+3\,B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^4+12\,m^3\,n+4\,m^3+49\,m^2\,n^2+36\,m^2\,n+6\,m^2+78\,m\,n^3+98\,m\,n^2+36\,m\,n+4\,m+40\,n^4+78\,n^3+49\,n^2+12\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^4+14\,m^3\,n+4\,m^3+71\,m^2\,n^2+42\,m^2\,n+6\,m^2+154\,m\,n^3+142\,m\,n^2+42\,m\,n+4\,m+120\,n^4+154\,n^3+71\,n^2+14\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {B\,b^3\,d\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(A + B*x^n)*(a + b*x^n)^3*(c + d*x^n),x)

[Out]

(A*a^3*c*x*(e*x)^m)/(m + 1) + (b^2*x*x^(4*n)*(e*x)^m*(A*b*d + 3*B*a*d + B*b*c)*(4*m + 11*n + 33*m*n + 82*m*n^2
+ 33*m^2*n + 61*m*n^3 + 11*m^3*n + 6*m^2 + 4*m^3 + m^4 + 41*n^2 + 61*n^3 + 30*n^4 + 41*m^2*n^2 + 1))/(5*m + 1
5*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 +
m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (a*x*x^(2*n)*(e*x)^
m*(3*A*b^2*c + B*a^2*d + 3*A*a*b*d + 3*B*a*b*c)*(4*m + 13*n + 39*m*n + 118*m*n^2 + 39*m^2*n + 107*m*n^3 + 13*m
^3*n + 6*m^2 + 4*m^3 + m^4 + 59*n^2 + 107*n^3 + 60*n^4 + 59*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 9
0*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274
*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (b*x*x^(3*n)*(e*x)^m*(A*b^2*c + 3*B*a^2*d + 3*A
*a*b*d + 3*B*a*b*c)*(4*m + 12*n + 36*m*n + 98*m*n^2 + 36*m^2*n + 78*m*n^3 + 12*m^3*n + 6*m^2 + 4*m^3 + m^4 + 4
9*n^2 + 78*n^3 + 40*n^4 + 49*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n
+ 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 +
225*m^2*n^3 + 85*m^3*n^2 + 1) + (a^2*x*x^n*(e*x)^m*(A*a*d + 3*A*b*c + B*a*c)*(4*m + 14*n + 42*m*n + 142*m*n^2
+ 42*m^2*n + 154*m*n^3 + 14*m^3*n + 6*m^2 + 4*m^3 + m^4 + 71*n^2 + 154*n^3 + 120*n^4 + 71*m^2*n^2 + 1))/(5*m +
15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4
+ m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (B*b^3*d*x*x^(5*n
)*(e*x)^m*(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50
*n^3 + 24*n^4 + 35*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^
4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^
3 + 85*m^3*n^2 + 1)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*x**n)**3*(A+B*x**n)*(c+d*x**n),x)

[Out]

Timed out

________________________________________________________________________________________